Rectangular channel cross sections

4.2. Rectangular channel cross sections

a) b)

Figure 4.11.: (a) The y-component of the lift force experienced by a particle located on the y axis as a function of its position for different aspect ratios w/h. (b) The x-component of the lift force for a particle positioned on the x axis. Both plots share the channel Reynolds number Re = 10 and particle size a = 0.4w.

equilibrium positions become destabilized. In experiments, the strong pressures necessary for inertial microfluidics cause considerable deformations [159]. We perform independent MPCD simulations with parameters P2 given in appendix E to compare the equilibrium positions. We plot the profile of the lateral probability density of the particle for a/w= 0.5,Re = 20 in Fig. 4.10b. In the probability density, stable equilibrium positions result in maxima and the unstable manifold of the saddle nodes results in areas with increased probability. The results agree with those of the LBM simulation. The main axes have saddle node stability, while the equilibrium positions on the diagonals are stable. Further, the distanced_eq ≈0.45 of the stable equilibrium positions from the channel center agrees with the value obtained by the LBM simulations.

ra-4. Channel geometry and inertial focusing

tios w/hin Fig. 4.11a. The symmetry of the system prevents any lift force in xdirection.

However, all positions on the y axes are unstable against perturbations in x direction and there are no stable equilibrium position on the yaxis. For aspect ratiosw/h >0.46, the lift force shows an unstable equilibrium position at the channel center and another equilibrium position located closer to the channel wall. The second equilibrium position is stable along they direction, but unstable along the xdirection. Therefore, it has sad-dle node stability. Increasing channel height, weakens the lift force close to the channel center. For intermediate aspect ratios w/h ≈ 0.46, the channel center becomes stable against perturbation in the y direction. In the full cross section, it is a saddle node, since it is still unstable along the xdirection. The change of stability of the channel cen-ter coincides with the change of the oucen-ter equilibrium position to a saddle node and the emergence of an additional unstable equilibrium position between the channel center and the outer equilibrium position. Further decreasing the aspect ratio w/h drives the lift force to be negative for all y values and the channel center remains the only equilibrium position on the y axis.

The lift force for particles located on the x axis (Fig. 4.11b) shows two equilibrium positions for all aspect ratios. The channel center is unstable against perturbation along the x axis and the outer equilibrium position is always stable against perturbation in x direction. We discuss below the change of stability along theydirection for both equilib-rium positions with varying aspect ratio. With decreasing aspect ratio, the magnitude of the lift force decreases. For the smallest aspect ratios (w/h≤0.33), the lift force profile develops a constant shape. Then, the distance to the short channel faces is large and the flow in the channel center plane resembles a plane Poiseuille flow.

The equilibrium positions and their stability (Fig. 4.12a) characterize the bifurcation of the lift force profile. Although we only plot the equilibrium positions on theyaxis, the equilibrium positions on the x axis for w/h >1 correspond to the equilibrium positions on theyaxis forw/h <1 and vice versa. First, we consider aspect ratios below unity, for which we show the lift forces in Fig. 4.11a. The outer equilibrium position shifts closer towards the channel wall as we decrease the aspect ratio. In the left inset of Fig. 4.12a, we illustrate the system for w/h ≈ 0.75. For an aspect ratio w/h ≈ 0.48, a subcritical pitchfork bifurcation occurs in which the channel center becomes a saddle node and an additional unstable equilibrium position appears. This bifurcation marks the point where the y = 0 plane becomes stable and the dynamics effectively two-dimensional.

By further decreasing the aspect ratio, the newly created unstable equilibrium position moves towards the channel wall until it merges with the outer equilibrium position and both vanish in a saddle node bifurcation. For aspect ratios above unity, the equilibrium positions correspond to the force curves shown in Fig. 4.11b. Here, the position of the outer equilibrium position changes only little. At about x/w > 1.33, the outer equilibrium position becomes stable, while the diagonal equilibrium positions vanish.

We illustrate the full cross section for this situation in the right inset.

For smaller particles, the bifurcation scenario changes. We plot the inertial lift force

82

4.2. Rectangular channel cross sections

a) b)

Figure 4.12.: (a) Equilibrium positionsyeq on theyaxis as a function of aspect ratiow/h for a particle with radius a = 0.4w. Here, we show the full stability along both x and y direction. The insets show typical equilibrium positions in a channel for w < h (left) and w > h (right). The plot corresponds to the lift forces shown in Fig. 4.11a. (b) The y component of the lift force as a function of particle position on theyaxis for a smaller particle with radius a = 0.2w. The inset mirrors the main plot in (a). It shows the equilibrium positions yeq of the lift force in the main plot as a function of aspect ratio w/h. Both plots show data for channel Reynolds number Re = 10.

along the y direction for a particle with radius a= 0.2w in Fig. 4.12b. The inset shows the corresponding bifurcation diagram. The lift force undergoes the subcritical pitchfork bifurcation for w/h ≈ 0.45. However, it does not show the saddle node bifurcation for low aspect ratios. This difference is also visible in the qualitative difference between the lift force profiles for large particles (Fig. 4.9c) and small particles (Fig.5.3a). Specifically, the lift force profile for smaller particles keeps a constant shape close to the channel wall for w/h≤0.25. For the channels with small aspect ratio the undisturbed flow field show significant shear gradients only close to the channel wall. However, only the small particle is able to explore these regions, where we expect a lift force according to the dynamic pressure model (Sect. 2.4.5). Furthermore, the outer equilibrium positions is stable for w/h= 1 and becomes unstable only for aspect ratios below w/h < 0.9. This change in stability coincides with the vanishing diagonal equilibrium positions.

The dynamic pressure model, introduced in Sect. 2.4.5, explains the features of the lift force profiles in term of the unperturbed flow profiles [20, 95]. The change in flow profile along the y axis (Fig. 3.10b) mirrors the change in lift force (Fig. 4.11a). With increasing aspect ratio the flow profile develops a blunted shape close to the channel axis, with appreciable gradients visible only close to the channel wall. In particular, for aspect ratios w/h ≤ 0.5, the flow profile changes drastically and the lift force profile undergoes the bifurcation to an effectively two-dimensional system. In contrast, on the short channel axis, the small change of the flow profile (Fig. 3.10a) mirrors the small

4. Channel geometry and inertial focusing

Figure 4.13.: The minimum height h2D required for a two-dimensional particle behavior as a function of Reynolds number for different particle sizes. The aspect ratio w/h2D

corresponds to the pitchfork bifurcation in the main plot of Fig. 4.12a and the inset of Fig. 4.12b.

change in the lift forces (Fig. 4.11b).

Devices using optical methods often require a well defined focal plane such that par-ticles always move in the focus of the optical instruments (for example [12]). Therefore, it is important to know the aspect ratio for which the system becomes effectively two-dimensional and particles move in they= 0 plane. In the previous sections we found that the transition coincides with the change of stability of the channel center from unstable to saddle node. We plot the minimum channel height h2D required for this transition in Fig. 4.13. The required height decreases for increasing particle size. Furthermore, the required height is a monotonically decreasing function of Reynolds number for all parti-cles but the smallest (a= 0.2w). For all investigated parameters, an aspect ratio below w/h= 0.43 is sufficient to render the system effectively two-dimensional. In experiments [12], the channel center plane was only for intermediate Reynolds numbers stable. In particular, for particle Reynolds numbers Rep = (a/w)²Re<5, particles collected in the channel center plane consistent with our results. Increased flow rates however destabi-lize the channel center plane. Our results do not explain these findings. However, the experiment [12] used a dense suspension of particles with small inter-particle distances.

Our simulations do not include the hydrodynamic interactions between particles. We therefore suspect that the missing hydrodynamic interactions explain the discrepancy.